On Beurling's uncertainty principle

We generalise a result of Hedenmalm to show that if a function $f$ on $\mathbb{R}$ is such that $\int_{$\mathbb{R}$^2} \bigl|f(x) , \hat f(y)\bigr| ,e^{\lambda \left|xy\right|} ,dx,dy = O( (1-\lambda)^{-N} )$as$\lambda \to 1-$,then$f$ is the product of a polynomial and a gaussian.